\section{1.8} 
\begin{frame}[allowframebreaks]{1.8. }

\vspace{-0.4cm}

1.8. The de Rham complex. 

We let $\Omega_X^*$ be the sheaf of germs of regular exterior differential forms on $X$. 

Since a $\mathcal{D}$-module may be viewed as a $\mathcal{O}_X$-module endowed with a flat connection, we may again form as usual (differential graded) sheaf of germs of exterior differential forms with coefficients in $M$,
\[
\Omega_X^*(M) = \Omega_X^* \otimes_{\mathcal{O}_X} M.
\]

Locally, with a choice of local coordinates $(x_i)$, its differential can be expressed as
\[
d(\omega \cdot m) = d\omega \cdot m + \sum_i dx_i \wedge \omega \cdot \partial_i m.
\]

It is easily checked that it is independent of the choice of local coordinates and, using the flatness of the connection, that $d^2 = 0$.

\end{frame}

